A265145 Number of lambda-parking functions of the unique strict partition lambda with parts i_1

1, 1, 2, 3, 3, 5, 4, 16, 8, 7, 5, 25, 6, 9, 12, 125, 7, 34, 8, 34, 16, 11, 9, 189, 15, 13, 50, 43, 10, 49, 11, 1296, 20, 15, 21, 243, 12, 17, 24, 253, 13, 64, 14, 52, 74, 19, 15, 1921, 24, 58, 28, 61, 16, 307, 27, 317, 32, 21, 17, 343, 18, 23, 98, 16807, 33

Offset: 1

Author

Alois P. Heinz, Dec 02 2015

Keywords

• nonn

Comments

A strict partition is a partition into distinct parts.

Examples

			n = 10 = 2*5 = prime(1)*prime(3) encodes strict partition [1,4] having seven lambda-parking functions: [1,1], [1,2], [2,1], [1,3], [3,1], [1,4], [4,1], thus a(10) = 7.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000
• Richard P. Stanley, Parking Functions, 2011.

Crossrefs

Cf. A000009, A000040, A265144, A265146, A265208.

Programs

Maple

p:= l-> (n-> n!*LinearAlgebra[Determinant](Matrix(n, (i, j)
         -> (t->`if`(t<0, 0, l[i]^t/t!))(j-i+1))))(nops(l)):
a:= n-> p((l-> [seq(l[j]+j-1, j=1..nops(l))])(sort([seq(
         numtheory[pi](i[1])$i[2], i=ifactors(n)[2])]))):
seq(a(n), n=1..100);

Mathematica

p[l_] := Function [n, n! Det[Table[Function[t, If[t<0, 0,
     l[[i]]^t/t!]][j-i+1], {i, n}, {j, n}]]][Length[l]];
a[n_] := If[n==1, 1, p[Function[l, Flatten[Table[l[[j]]+j-1,
     {j, 1, Length[l]}]]][Sort[Flatten[Table[Table[PrimePi[
     i[[1]]], {i[[2]]}], {i, FactorInteger[n]}]]]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Aug 21 2021, after Alois P. Heinz *)